Length, Area, and Volume - Outcomes

Transcription

1 1 Length, Area, and Volume - Outcomes Solve problems about the perimeter and area of triangles, rectangles, squares, parallelograms, trapeziums, discs, sectors, and figures made from combinations of these. Estimate the area of regular and irregular shapes. Use the trapezoidal rule to approximate areas.

2 2 These formulas are NOT supplied in the F&T booklets Solve Problems about Rectangles Rectangles are four-sided shapes with four right angles. Their opposite sides are equal in length, usually called their width, w, and length, l. Formula: Perimeter = w + l + w + l = 2w + 2l. Formula: Area = lw

3 3 Solve Problems about Rectangles Find the area and perimeter of each of the following:

4 4 Solve Problems about Rectangles Find the missing side lengths for each of the following rectangles:

5 5 Solve Problems about Rectangles The diagram shows a sitting room. a) What is the area of the floor? b) If rpet osts per m 2, what would be the cost of carpeting the whole floor? c) If floorboards are 20 cm wide and 6 m long, how many would be required to cover the floor?

6 6 Solve Problems about Rectangles If each shape below has the same width and height, which has the greatest perimeter (assuming all right angles)?

7 FL P2 Q FL P2 Q1 Solve Problems about Rectangles A wire of length 3 metres is bent to form a rectangle, one side of which is 45 cm. Calculate the length of the longer side and hence calculate the area enclosed by the rectangle. The perimeter of a rectangular field is 280 m. The length of the longer side is 100 m. Find: i. the length of the shorter side ii. the area of the field

8 8 Solve Problems about Triangles Triangles are three-sided shapes with some sub-types: Equilateral triangles have three sides of equal length (also three angles of equal size, 60 ). Isosceles triangles have two sides of equal length (also two opposite angles of equal size). Scalene triangles have no sides of equal length (also no angles of equal size). Right-angled triangles contain a right angle and may also be either isosceles or scalene.

9 9 Perimeter formula not given in F&T booklet Solve Problems about Triangles For any triangle with sides a, b, and c: Formula: Perimeter = a + b + c Formula: Area = 1 ah, with h a. 2

10 10 Solve Problems about Triangles Find the perimeter of each of the following triangles:

11 11 Solve Problems about Triangles Find the area of each of the following triangles:

12 12 Solve Problems about Triangles Find the missing lengths in each of the following triangles:

13 FL P2 Q FL P1 Q1 Solve Problems about Triangles The area of triangle shown is 10 cm 2. The length of the base is 8 cm. Find h, the perpendicular height of the triangle. The area of triangle abc is cm 2. The length of [bc] is 8.1 cm. The height of the triangle is h cm. Calculate the value of h.

14 14 Perimeter formula not given in F&T booklet Solve Problems about Parallelograms Parallelograms are quadrilaterals whose opposite sides are parallel. Formula: P = 2a + 2b Formula: A = ah Parallelograms with a = b are called rhombuses Rectangles and squares are types of parallelogram

15 15 Solve Problems about Parallelograms Find the perimeter and area of each of the following parallelograms: a) b) c)

16 16 Solve Problems about Parallelogram Find the missing length in each of the following parallelograms: a) b) Area = 40 units 2 Area = 15 units 2

17 17 Solve Problems about Trapeziums Find the perimeter and area of each trapezium: a) b) c)

18 FL P2 Q2a Solve Problems about Trapeziums A trapezium abcd has the following dimensions: the length of [ab] is 8 cm, the length of [dc] is 12 cm, the height is 7 cm. Find the area of the trapezium.

19 19 Solve Problems about Discs The perimeter of a disc or circle is called its circumference. Two other numbers are important for discs and circles. The radius is the distance from the centre of the disc to the circumference. The diameter is the length of a line passing through the centre of the disc, ending at the circumference. diameter = 2 radius Circumference Diameter Radius

20 20 π Solve Problems about Discs The circumference of a circle is given by a formula: Circumference = 2 π radius C = 2πr e.g. Find the circumference of each of the circles below: 28 cm 3 m

21 21 Solve Problems about Discs The area of a circle is given by a formula: Area = π radius 2 A = πr 2 e.g. Find the area of each of the circles below: 28 cm 3 m

22 22 Solve Problems about Discs If we only have part of a circle, we only have part of the circumference. There are usually other sides to add to the perimeter. Find the perimeter of the shapes below. a) b)

23 23 Solve Problems about Discs We can generalise the perimeter of any sector (like a slice of pizza or cake). l = 2πr θ 360 l = arclength r = radius θ = angle at centre Total perimeter of a sector is given by: P = l + 2r

24 24 Solve Problems about Discs We can generalise the area of any sector (like a slice of pizza or cake). A = πr 2 A = area r = radius θ 360 θ = angle at centre

25 25 l = 2πr A = πr 2 θ 360 θ 360 Solve Problems about Discs Find the perimeter of each of the following shapes: a) b) c)

26 26 l = 2πr A = πr 2 θ 360 θ 360 Solve Problems about Discs Find the missing number in each of the following shapes: a) b) c)

27 27 l = 2πr A = πr 2 θ 360 θ 360 Area of Disks If we only have part of a circle, we only have part of the area. Find the area of the shapes below. a) b)

28 28 l = 2πr A = πr 2 θ 360 θ 360 Solve Problems about Discs Find the area of each of the following shapes: a) b) c)

29 29 l = 2πr A = πr 2 θ 360 θ 360 Solve Problems about Discs Find the missing number in each of the following shapes: a) b) c)

30 30 SP about Combined Shapes Find the shaded area of the following shapes: a) b) c)

31 31 AP about Combined Shapes Find the total perimeter and area of the following shapes: a) b) c)

32 FL Q5a Solve Problems about Discs A circular disc is cut out of a square piece of metal as shown below. The disc has a radius of 3 cm. i. Find the area of the disc. Give your answer correct to 1 decimal place. ii. Find the area of metal left after the disc has been removed. Give your answer in cm 2, correct to 1 decimal place.

33 FL Q10b Solve Problems about Discs T e i gr elow s ows t e logo for Ger l ine s company, TLC Ltd. Each letter is contained in a quarter circle of radius 6 cm. i. Find the length of the perimeter of this logo correct to the nearest whole number. ii. iii. Find the area of the logo correct to three decimal places. The material used to construct t e logo osts per 2. Find the cost of constructing 50 logos, correct to the nearest euro.

34 34 Approximate Area with the Trapezoidal Rule Consider an irregular shape:

35 35 Approximate Area with the Trapezoidal Rule Divide it up into strips of equal width:

36 36 Approximate Area with the Trapezoidal Rule Approximate these strips with trapeziums:

37 37 Approximate Area with the Trapezoidal Rule Use the trapezoidal rule formula to approximate the area: Formula: A h 2 y 1 + y n + 2 y 2 + y 3 + y y n 1

38 OL P2 Q1b Approximate Area with the Trapezoidal Rule The sketch shows a section of a wall that is to be painted. At equal intervals of 1.2 m along the bottom of the wall, perpendicular measurements are made to the uneven edge, as shown on the sketch.

39 39 continued Approximate Area with the Trapezoidal Rule i. Use the trapezoidal rule to estimate the area of the section of the wall. ii. How many litres of paint are required to paint the section of the wall, if 1 litre of paint covers an area of 2.2 m 2? Give your answer correct to the nearest litre.

40 OL P2 Q5b Approximate Area with the Trapezoidal Rule The diagram shows a sketch of a field ABCD that has one uneven edge. At equal intervals of 5 m along [BC], perpendicular measurements are made to the uneven edge, as shown on the sketch.

41 41 continued Approximate Area with the Trapezoidal Rule i. Use the trapezoidal rule to estimate the area of the field. ii. The actual area of the field is 200 m 2. Find the percentage error in the estimate.